
This is a decoupled system of BSDE.
By Theorem 4.1 and Theorem 5.2 of [1]}, there exists a unique solution \((\underline{P}(i),\underline{\Lambda }(i))\in L^\infty _{\mathcal {F}^W}(0, N; \mathbb {R})\times L^{2}_{\mathcal {F}^{W}}(0, N;\mathbb {R}^{n})\)  with \(\underline{P}(i)\ge 0 \)  for all \(i\in \mathcal {M}\) .
